Integrating a Models and Modeling Perspective With Existing Research and Practice

This chapter provides an illustration of how the models and modeling perspective can be used in collegiate mathematics education research and instruction. The models and modeling approach (chapter 1) provides instructional designers with a well-defined structure for creating curriculum. The curricular activities for this approach, referred to as model-eliciting activities (Lesh, Hoover, Hole, Kelly, & Post, 2000), are designed to encourage students to make sense of meaningful situations, and to invent, extend, and refine their own mathematical constructs. The resultant student products reveal students' thinking and provide both teachers and researchers with a powerful lens for viewing students' reasoning and concept development. In this chapter, we discuss how this perspective has influenced both our research and instruction. For the past 5 years we have been engaged in research to investigate undergraduate students' understanding of rate of change (Carlson, 1998) and covariational reasoning (coordinating two varying quantities while attending to the ways in which they change in relation to each other; Carlson, Jacobs, & Larsen, 2001). These studies have identified aspects of covariational reasoning and have pointed to specific difficulties that students encounter when reasoning about dynamic events. Much of the data from our past research was gathered using specific mathematical tasks. Using the insights gained from this earlier research (Carlson, 1998), we modified these tasks to adhere to the six principles for developing model-eliciting activities (Lesh et al., 2000). We then utilized these new activities in a small-scale study that investigated undergraduate students' covariational reasoning abilities. First, we present a brief sketch of Marilyn Carlson's research on covariational reasoning and the theoretical framework that resulted from that work. We then describe the process of converting a specific covariation task, discussed in Carlson (1998) to a modeling eliciting activity. This is followed by a description of a study that uses the newly developed model-eliciting activities.